Theorem curry1val 8146

Description: The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypothesis

Ref	Expression
curry1.1	⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))

Assertion

Ref	Expression
curry1val	⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐺‘𝐷) = (𝐶𝐹𝐷))

Proof of Theorem curry1val

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		curry1.1	. . . 4 ⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))
2	1	curry1 8145	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)))
3	2	fveq1d 6913	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐺‘𝐷) = ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷))
4		eqid 2763	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))
5	4	fvmptndm 7051	. . . . . 6 ⊢ (¬ 𝐷 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = ∅)
6	5	adantl 487	. . . . 5 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐷 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = ∅)
7		fndm 6667	. . . . . . 7 ⊢ (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
8	7	adantr 486	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → dom 𝐹 = (𝐴 × 𝐵))
9		simpr 490	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐷 ∈ 𝐵)
10	9	con3i 155	. . . . . 6 ⊢ (¬ 𝐷 ∈ 𝐵 → ¬ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))
11		ndmovg 7631	. . . . . 6 ⊢ ((dom 𝐹 = (𝐴 × 𝐵) ∧ ¬ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (𝐶𝐹𝐷) = ∅)
12	8, 10, 11	syl2an 608	. . . . 5 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐷 ∈ 𝐵) → (𝐶𝐹𝐷) = ∅)
13	6, 12	eqtr4d 2801	. . . 4 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐷 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))
14	13	ex 418	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (¬ 𝐷 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷)))
15		oveq2 7454	. . . 4 ⊢ (𝑥 = 𝐷 → (𝐶𝐹𝑥) = (𝐶𝐹𝐷))
16		ovex 7479	. . . 4 ⊢ (𝐶𝐹𝐷) ∈ V
17	15, 4, 16	fvmpt 7019	. . 3 ⊢ (𝐷 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))
18	14, 17	pm2.61d2 183	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))
19	3, 18	eqtrd 2798	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐺‘𝐷) = (𝐶𝐹𝐷))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ∅c0 4281  {csn 4588   ↦ cmpt 5190   × cxp 5654  ^◡ccnv 5655  dom cdm 5656   ↾ cres 5658   ∘ ccom 5660   Fn wfn 6556  ‘cfv 6561  (class class class)co 7446  2^nd c2nd 8027

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-1st 8028  df-2nd 8029

This theorem is used by:  nvinvfval  31275  hhssabloilem  31896